On the zone complexity of a vertex

TL;DR

This paper proves that in any arrangement of n lines in the real projective plane, there exists a vertex whose local face complexity reduces to at most five when removing the two lines passing through it.

Contribution

It establishes a new bound on the local face complexity around a vertex in line arrangements, advancing understanding of geometric arrangements.

Findings

Existence of a vertex with face size at most 5 after removing two lines

Improves understanding of local face structures in line arrangements

Provides a bound relevant for combinatorial geometry studies

Abstract

Let $L$ be a set of $n$ lines in the real projective plane in general position. We show that there exists a vertex $v\in \A(L)$ such that $v$ is positioned in a face of size at most 5 in the arrangement obtained by removing the two lines passing through $v$.